THE Z-TRANSFORM ANALYSIS OF LINEAR SAMPLED-DATA SYSTEMS 959 
g(t). The constant c is so chosen that all the poles of F(p) lie to the left 
of the imaginary axis displaced by c. 
The complex-convolution integral can be applied to the case where 
the two time functions involved are those given in (4.6). In this case, 
F(p) is the Laplace transform of the function f(t) being sampled. The 
Laplace transform of the impulse sequence 67(¢) is given by the following: 
++ 2 
sm) Ss > e-nT(s—p) (4.8) 
n=0 
This infinite summation can be expressed in closed form by using the 
formula for the sum of a geometric series. The expression (4.8) is seen 
to be an infinite geometric progression with a ratio of e-7°-»), Thus, 
G(s — p) can be written 
1 
G(s = Dp) — 1 — e-fG=>) (4.9) 
The integral giving the Laplace transform of f*(¢) can thus be expressed 
in the following form 
e+jo 
off] = 5 i _ Oya ® (4.10) 
Evaluation of (4.10) is accomplished through contour integration by 
closing the path of integration along the imaginary axis to encompass 
either the left or right half planes and evaluating the residues at the 
various poles enclosed. It is noted that, for stable systems, all the poles 
of F(p) lie in the left half of the s plane, as shown in Fig. 4.2. On the 
other hand, the poles of the second term in the integrand of (4.10) are 
infinite in number, occurring at the complex frequency pn, for which the 
angle of the exponential term is m2z, where m is an integer. The con- 
dition for the angle of the exponential term to be a multiple of 27 is that 
.2rm 
Dm = 8 —j a (4.11) 
for all integral values of m. Thus the second function in (4.10) has an 
infinite number of poles separated by 727/T along the displaced imaginary 
axis, as shown in Fig. 4.2. The path of integration is along the imaginary 
axis displaced by the constant c and is shown in the figure with c so chosen 
that all the poles of F(p) lie to the left of this line. 
To apply the method of residues, a closed contour of integration is 
formed by closure either to the right or the left of c, resulting in the 
contours marked T; or Tz. It is seen that if closure is effected to the 
right, there will be an infinite number of poles enclosed, whereas if 
closure is effected to the left, only the finite number of poles contained 
